c
2006 by Institut Mittag-Leffler. All rights reserved
Classification of negatively pinched manifolds with amenable fundamental groups
by
Igor Belegradek
Georgia Institute of Technology Atlanta, GA, U.S.A.
Vitali Kapovitch
University of Maryland College Park, MD, U.S.A.
Contents
1. Introduction . . . 229
2. Sketch of the proof of (1)⇒(3) . . . 232
3. Topological digression . . . 233
4. Parallel transport through infinity and rotation homomorphism . 234 5. Passing to the limit . . . 235
6. Controlling injectivity radius . . . 238
7. Product structure at infinity . . . 243
8. Tubular neighborhood of an orbit . . . 246
9. The normal bundle is flat . . . 247
9.1. The normal bundle in a stratum is flat . . . 248
9.2. The normal bundle to a stratum is flat . . . 249
10. Infranilmanifolds are horosphere quotients . . . 251
11. On geometrically finite manifolds . . . 253
Appendix A. Lemmas on nilpotent groups . . . 254
Appendix B. Isometries are smooth . . . 255
Appendix C. Local formula of Ballmann and Br¨uning . . . 255
Appendix D. Concave functions and submetries on Alexandrov spaces 257 References . . . 258
1. Introduction
In this paper we study manifolds of the form X/Γ , where X is a simply-connected complete Riemannian manifold with sectional curvatures pinched (i.e. bounded) between two negative constants, and Γ is a discrete torsion-free subgroup of the isometry group of X. According to [10], if Γ is amenable, then either Γ stabilizes a biinfinite geodesic, or else Γ fixes a unique point z at infinity. The case when Γ stabilizes a biinfinite
230 i. belegradek and v. kapovitch
geodesic is completely understood, namely the normal exponential map to the geodesic is a Γ -equivariant diffeomorphism, hence X/Γ is a vector bundle over S1; there are only two such bundles each admitting a complete hyperbolic metric.
If Γ fixes a unique point z at infinity (such groups are calledparabolic), then Γ stabilizes horospheres centered at z and permutes geodesics asymptotic to z, so that, given a horosphereH, the manifold X/Γ is diffeomorphic to the product ofH/Γ withR. We refer toH/Γ as ahorosphere quotient. In this case, a delicate result of B. Bowditch [6]
shows that Γ must be finitely generated, which, by Margulis’ lemma [3], implies that Γ is virtually nilpotent.
The main result of this paper is a diffeomorphism classification of horosphere quo- tients, namely we show that, up to a diffeomorphism, the classes of horosphere quotients and (possibly noncompact) infranilmanifolds coincide.
By an infranilmanifold we mean the quotient of a simply-connected nilpotent Lie group G by the action of a torsion-free discrete subgroup Γ of the semidirect product of G with a compact subgroup of Aut(G) .
Theorem 1.1. For a smooth manifold N the following are equivalent:
(1) N is a horosphere quotient;
(2) N is diffeomorphic to an infranilmanifold;
(3) N is the total space of a flat Euclidean vector bundle over a compact infranil- manifold.
The implication (3)⇒(2) is straightforward; (2)⇒(1) is proved by constructing an explicit warped product metric of pinched negative curvature. The proof of the implication (1)⇒(3) occupies most of the paper, and depends on the collapsing theory of J. Cheeger, K. Fukaya and M. Gromov [14].
If N is compact (in which case the conditions (2) and (3) are identical), then the implication (1)⇒(2) follows from Gromov’s classification of almost flat manifolds, as improved by E. Ruh, while the implication (2)⇒(1) is new. If N is noncompact, then Theorem 1.1 is nontrivial even when π1(N)∼=Z, although the proof does simplify in this case. A direct algebraic proof of (2)⇒(3) was given in [39, Theorem 6], but the case when N is a nilmanifold was already treated in [25], where it is shown that any nilmanifold is diffeomorphic to the product of a compact nilmanifold and a Euclidean space.
We postpone the discussion of the proof till §2, and just mention that the proof also gives geometric information about horosphere quotients, e.g. we show that H/Γ is diffeomorphic to a tubular neighborhood of some orbit of an N-structure on H/Γ .
By Chern–Weil theory, any flat Euclidean vector bundle has zero rational Euler and Pontryagin classes. Moreover, by [38], any flat Euclidean bundle with virtually abelian
holonomy is isomorphic to a bundle with finite structure group. Thus the vector bundle in (3) becomes trivial in a finite cover, and has zero rational Euler and Pontryagin classes, and in particular, any horosphere quotient is finitely covered by the product of a compact nilmanifold and a Euclidean space.
Corollary 1.2. A smooth manifold M with amenable fundamental group admits a complete metric of pinched negative curvature if and only if it is diffeomorphic to the M¨obius band, or to the product of a line and the total space of a flat Euclidean vector bundle over a compact infranilmanifold.
The pinched negative curvature assumption in Corollary 1.2 cannot be relaxed to
−16sec60 or sec6−1 , e.g. because these assumptions do not force the fundamental group to be virtually nilpotent [6, §6]. More delicate examples come from the work of M. Anderson [1], who proved that each vector bundle over a closed nonpositively curved manifold (e.g. a torus) carries a complete Riemannian metric with −16sec60 . Since in each dimension there are only finitely many isomorphism classes of flat Euclidean bundles over a given compact manifold, all but finitely many vector bundles over tori admit no metrics of pinched negative curvature. Also−16sec(M)60 can be turned into sec(M×R)6−1 for the warped product metric on M×R with warping function et [5], hence Anderson’s examples carry metrics with sec6−1 after taking product with R.
Specifically, ifE is the total space of a vector bundle over a torus with nontrivial rational Pontryagin class, thenM=E×Rcarries a complete metric of sec6−1 but not of pinched negative curvature. Finally, Anderson also showed that every vector bundle over a closed negatively curved manifold admits a complete Riemannian metric of pinched negative curvature, hence amenability of the fundamental group is indispensable.
Because an infranilmanifold with virtually abelian fundamental group is flat, Theo- rem 1.1 immediately implies the following result.
Corollary 1.3. Let M be a smooth manifold with virtually abelian fundamental group. Then the following are equivalent:
(1) M admits a complete metric of sec≡−1 ;
(2) M admits a complete metric of pinched negative curvature.
In [7] Bowditch developed several equivalent definitions of geometrical finiteness for pinched negatively curved manifolds, and conjectured the following result.
Corollary 1.4. Any geometrically finite pinched negatively curved manifold X/Γ is diffeomorphic to the interior of a compact manifold with boundary.
We believe that the main results of this paper, including Corollary 1.4, should extend to the orbifold case, i.e. when Γ is not assumed to be torsion free. However, working
232 i. belegradek and v. kapovitch
in the orbifold category creates various technical difficulties, both mathematical and expository, and we do not attempt to treat the orbifold case in this paper.
Acknowledgements. It is a pleasure to thank Ilya Kapovich and Derek J. S. Robinson for Lemma A.1, Hermann Karcher for pointing us to [8], Robion C. Kirby for Lemma 7.3, Anton Petrunin for numerous helpful conversations and suggestions and particularly for Lemma D.1, and Xiaochun Rong and Kenji Fukaya for helpful discussions on collaps- ing. We are grateful to the referee for various editorial comments. This work was partially supported by the NSF grants DMS-0352576 (Belegradek) and DMS-0204187 (Kapovitch).
2. Sketch of the proof of (1)⇒(3)
The Busemann function corresponding to z gives rise to a C2-Riemannian submersion X/Γ!R whose fibers are horosphere quotients each equipped with the induced C1- Riemannian metricgt. By the Rauch comparison theorem, the second fundamental form of a horosphere is bounded in terms of curvature bounds ofX (cf. [8]). In particular, each fiber has curvature uniformly bounded above and below in comparison sense. Letσ(t) be a horizontal geodesic in X/Γ , i.e. a geodesic that projects isometrically to R. Because of the exponential convergence of geodesics in X, the manifold X/Γ is “collapsing” in the sense that the unit balls around σ(t)∈X/Γ form an exhaustion of X/Γ and have small injectivity radius for large t. Similarly, each fiber of X/Γ!R also collapses, and in fact X/Γ is noncollapsed in the direction transverse to the fibers.
There are essential difficulties in applying the collapsing theory of [14] toX/Γ . First, we do not know whether (X/Γ, σ(t)) converges in pointed Gromov–Hausdorff topology to a single limit space. By general theory, the family (X/Γ, σ(t)) is precompact and thus has many converging subsequences. While different limits might be nonisometric, one of the main steps of the proof is obtaining a uniform (i.e. independent of the subsequence) lower bound on the “injectivity radius” of the limit spaces at the base point. This is done by a comparison argument involving taking “almost square roots” of elements of Γ , and using the flat connection of [6] discussed below. Another complication is that the N- structure on (X/Γ, σ(t)) provided by [14] may well have zero-dimensional orbits outside the unit ball around σ(t) , in other words a large noncompact region of (X/Γ, σ(t)) may be noncollapsed, which makes it hard to control topology of the region.
However, once the “injectivity radius” bound is established, critical points of distance functions considerations yield the “product structure at infinity” for X/Γ , and also for (H/Γ, gt) if t is large enough.
Furthermore, one can show that H/Γ is diffeomorphic to the normal bundle of an orbit Ot of the N-structure. The orbit corresponds to the point in a limit space given to us by the convergence, and at which we get an “injectivity radius” bound. This depends on a few results on Alexandrov spaces with curvature bounded below, with key ingredients provided by [21] and [31].
By the collapsing theory, the structure group of the normal bundle to the orbit of an N-structure is a finite extension of a torus group [14]. Of course, not every such a bundle has a flat Euclidean structure.
The flatness of the normal bundle to the orbit is proved using a remarkable flat connection discovered by B. Bowditch [6], and later in a different disguise by W. Ball- mann and J. Br¨uning [2], who were apparently unaware of [6]. It follows from [6] or [2]
that each pinched negatively curved manifold X/Γ , where Γ fixes a unique point z at infinity, admits a natural flat C0-connection that is compatible with the metric and has nonzero torsion, and such that on short loops it is close to the Levi–Civita connection.
Furthermore, the parallel transport of the connection preserves the fibration of X/Γ by horosphere quotients. Hence each horosphere quotient has flat tangent bundle.
In fact, we prove a finer result that the normal bundle to Ot is also flat, for suit- able large t. (Purely topological considerations are useless here since there exist vector bundles without flat Euclidean structure whose total spaces have flat Euclidean tangent bundles, for example this happens for any nontrivial orientable R2-bundle over the 2 - torus that has even Euler number.) It turns out that Ot sits with flat normal bundle in a totally geodesic stratum of the N-structure, so it suffices to show that the normal bundle to the stratum is flat, when restricted toOt. Now, since the above flat connection is close to the Levi–Civita connection, the normal bundle is “almost flat”, and it can be made flat by averaging via center of mass. This completes the proof.
Throughout the proof we use the collapsing theory developed in [14]. This paper is based on the earlier extensive work of Fukaya, and Cheeger–Gromov, and many argu- ments in [14] are merely sketched. We suggest reading [17] for a snapshot of the state of affairs before [14], and [34], [33], [16] for a current point of view.
3. Topological digression
The result of Bowditch [6] that horosphere quotients have finitely generated fundamental groups actually implies that any horosphere quotient is homotopy equivalent to a compact infranilmanifold (because any torsion-free finitely generated virtually nilpotent group is the fundamental group of a compact infranilmanifold [15], and because for aspherical manifolds any π1-isomorphism is induced by a homotopy equivalence).
234 i. belegradek and v. kapovitch
To help appreciate the difference between this statement and Theorem 1.1, we discuss several types of examples that are allowed by Bowditch’s result, and are ruled out by Theorem 1.1. The simplest example is a vector bundle over an infranilmanifold with nonzero rational Euler or Pontryagin class: such a manifold cannot be the total space of a flat Euclidean bundle, as is easy to see using the fact that the tangent bundle to any infranilmanifold is flat.
Another example is the product of a closed infranilmanifold and a contractible man- ifold of dimension >2 that is not simply-connected at infinity. Finally, even more so- phisticated examples come from the fact that below metastable range (starting at which any homotopy equivalence is homotopic to a smooth embedding, by Haefliger’s embed- ding theorem) there are many smooth manifolds that are thickenings of say a torus, yet are not vector bundles over the torus. It would be interesting to see whether the
“weird” topological constructions of this paragraph can be realized geometrically, even as nonpositively curved manifolds.
4. Parallel transport through infinity and rotation homomorphism LetX be a simply-connected complete pinched negatively curvedn-manifold normalized so that −a26sec(X)6−1 . One of the key properties of X used in this section is that any two geodesic rays in X that are asymptotic to the same point at infinity converge exponentially, i.e. for any asymptotic raysγ1(t) and γ2(t) , withγ1(0) andγ2(0) lying on the same horosphere, the function d(γ1(t), γ2(t)) is monotonically decreasing as t!∞, and
e−atc1(a, d(γ1(0), γ2(0)))6d(γ1(t), γ2(t))6e−tc2(a, d(γ1(0), γ2(0))),
where ci(a, d) is linear in d for small d. This is proved by triangle comparison with spaces of constant negative curvature.
Bowditch introduced a connection onX that we now describe (see [6,§3] for details).
Fix a pointz at infinity ofX. Letwi!zasi!∞. For anyx, y∈X, consider the parallel transport map from x to wi followed by the parallel transport from wi to y along the shortest geodesics. This defines an isometry between the tangent spaces at x and y. By [6, Lemma 3.1], this map converges to a well-defined limit isometry Pxy∞:TxM!TyM as i!∞. We refer to Pxy∞ as theparallel transport through infinity from x to y.
We denote the Levi–Civita parallel transport fromxtoy along the shortest geodesic by Pxy; clearly, if x and y lie on a geodesic ray that ends at z, then Pxy∞=Pxy. A key feature ofP∞ is that it approximates the Levi–Civita parallel transport on short geodesic segments (see [6, Lemma 3.2]; more details can be found in [11,§6]). This is because any geodesic triangle in X spans a “ruled” surface of area at most the area of the comparison
triangle in the hyperbolic plane of sec=−1 . By exponential convergence of geodesics, the area of the comparison triangle is bounded above by a constant times the shortest side of the triangle. As the holonomy around the circumference of the triangle is bounded by the integral of the curvature over its interior, we conclude that |Pxy−Pxy∞|6q(a)d(xy) , where q(a) is a constant depending only on a.
Given x∈X, fix an isometry Rn!TxX and translate it around X using P∞. This defines a P∞-invariant trivialization of the tangent bundle to X. Let Isoz(X) be the group of isometries of X that fixes z. For any point y∈X look at the map Isoz(X)!O(n) given by γ7!Pγ(y)y∞ dγ. It turns out that this map is a homomorphism independent ofy. We call it therotation homomorphism. Starting with a different base point x∈X or a different isometry Rn!TxX has the effect of replacing the rotation homomorphism by its conjugate.
Now, if Γ is a discrete torsion-free subgroup of Isoz(X) , then, since the rotation homomorphism is independent of y, P∞ gives rise to a flat connection on X/Γ with holonomy given by the rotation homomorphism. By the above discussion of P∞, this is a C0-flat connection that is compatible with the metric and close to the Levi–Civita connection on short loops. Of course, this connection has torsion.
Remark 4.1. The above discussion is easily seen to be valid if X is a simply- connected completeC1-Riemannian manifold of pinched negative curvature in the com- parison sense. This is because any such C1-metric can be approximated uniformly in C1-topology by smooth Riemannian metrics of pinched negative curvature [28], perhaps with slightly larger pinching. Then the distance functions and Levi–Civita connections converge uniformly in C0-topology, and we recover all the statements above.
Remark 4.2. The connection of Bowditch, that was described above, was reinvented later in a different disguise by Ballmann and Br¨uning [2, §3]. The connection in [2] is defined by an explicit local formula in terms of the curvature tensor and the Levi–Civita connection of X. Actually, [2] only discusses the case of compact horosphere quotients;
however, all the arguments there are local, hence they apply to any horosphere quotient.
The only feature which is special for compact horosphere quotients is that in that case the connection has finite holonomy group [2], as follows from estimates in [11]. For noncompact horosphere quotients, the holonomy need not be finite as seen by looking at a glide rotation with irrational angle inR3, thought of as a horosphere in the hyperbolic 4 -space. We never have to use [2] in this paper; however, for completeness, we discuss their construction in Appendix C, where we also show that the connections of [6] and [2]
coincide.
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5. Passing to the limit
LetX be a simply-connected complete pinched negatively curvedn-manifold normalized so that −a26sec(X)6−1 , let c(t) be a biinfinite geodesic in X, and let Γ be a closed subgroup of Iso(X) that fixes the pointc(∞) at infinity. We refer to the gradient flow bt of a Busemann function for c(t) asBusemann flow. Following Bowditch, we sometimes use the notation x+t:=bt(x) .
SinceX has bounded curvature and infinite injectivity radius, the family (X, c(t),Γ) has a subsequence (X, c(ti),Γ) that converges to (X∞, p, G) in the equivariant pointed C1,α-topology [32, Chapter 10]. Here X∞ is a smooth manifold with C1,α-Riemannian metric that has infinite injectivity radius and the same curvature bounds as X in the comparison sense, and G is a closed subgroup of Iso(X∞) . Note that Iso(X∞) is a Lie group that acts on X by C3-diffeomorphisms (this last fact is probably known but for a lack of reference we give a simple proof in Appendix B).
Furthermore, geodesic rays inX that start at uniformly bounded distance fromc(ti) converge to rays inX∞. In particular, the raysc(t+ti) starting atc(ti) converge to a ray c∞(t) in X∞ that starts atp, and the corresponding Busemann functions also converge.
Since the Busemann functions on X are C2 [19], they converge to a C1 Busemann function on X∞. Thus the horosphere passing through p is a C1-submanifold of X∞, and is the limit of horospheres passing through c(ti) . The Busemann flow is C1 on X, and C0 on X∞. Since the horospheres in X and X∞ have the same dimension, the sequence of horospheres passing throughc(ti) does not collapse, and, more generally, each horosphere centered at c∞(∞) is the limit of a noncollapsing sequence of horospheres in X.
It is easy to see that the group G fixes c∞(∞) , i.e. any γ∈G takes c∞ to a ray asymptotic to c∞. Furthermore, G leaves the horospheres corresponding to c∞(∞) invariant.
Thus, one can define the rotation homomorphism φ∞:G!O(n) corresponding to the point c∞(∞) . The point only determines φ∞ up to conjugacy, so we also need to fix an isometry L:Rn!TpX∞. Similarly, a choice of an isometry Li:Rn!Tc(ti)X specifies the rotation homomorphismφi: Γ!O(n) corresponding to the point c(∞) . We can assume that φi=φ0 for each i, by choosing Li equal to L0 followed by the parallel transport Pc(t∞
0),c(ti)=Pc(t0),c(ti). Henceforth we denote φ0 by φ. Also it is convenient to choose L as follows.
Lemma 5.1. After passing to a subsequence of (X, c(ti),Γ), there exists L such that if γi!γ, then φ(γi)!φ∞(γ).
Proof. Since (X, c(ti),Γ)!(X∞, p, G) in pointed equivariantC1,α-topology, we can
find the corresponding C1,α approximations fi:B(c(ti),1)!B(p,1) . We may assume that dfi:Tc(ti)X!TpX∞ is an isometry for all i. By compactness of O(n) , dfiLi
subconverge to an isometry L: Rn!TpX, so, by modifying fi slightly, we can assume that dfiLi=L. ThisL is then used to defineφ∞, and it remains to show that if γi!γ, then φ(γi)!φ∞(γ) . For the rest of the proof we suppress Li and L.
Since dγi!dγ, it is enough to show that parallel transports through infinity from c(ti) to γi(c(ti)) converge to the parallel transport through infinity from p to γ(p) .
For x∈X and c(t) that lie on the same horosphere, and for any s>t, denote by Px,s,c(t):Tc(t)M!TxM the parallel transport along the piecewise geodesic pathx, x+s, c(t)+s, c(t) . Here c(t)+s=c(t+s) and x+s also lie on the same horosphere. Now
|Pxc(t)∞ −Px,s,c(t)| can be estimated as
|Px+s,c(t)+s∞ −Px+s,c(t)+s|6q(a)d(x+s, c(t)+s)6q(a)e−sc2(a, d(x, c(t))).
By Remark 4.1, the same estimate holds for X∞ and c∞.
Fix ε>0 and pick R>0 such that d(c(ti), γi(c(ti)))6R for all i. Take a large enough s so that q(a)c2(a, R)e−s<ε. Since B(c(ti), R+s) converges to B(p, R+s) in C1,α-topology, and γi!γ, we conclude that Pγi(c(ti)),s,c(ti)!Pγ(p),s,p in C0-topology, or more formally,
|dfiPγi(c(ti)),s,c(ti)dγi−Pγ(p),s,pdγ|< ε
for large i, where fi is the C1,α-approximation. By the estimate in the previous para- graph, |Pγi(c(ti)),s,c(ti)−Pγ∞
i(c(ti))c(ti)|<ε and |Pγ(p),s,p−Pγ(p),p∞ |<ε, so the triangle in- equality implies that
|dfiPγ∞
i(c(ti))c(ti)dγi−Pγ(p),p∞ dγ|<3ε
for large i. Hence, |φ(γi)−φ∞(γ)|<3ε for all large i, and, since ε>0 is arbitrary, it follows that φ(γi)!φ∞(γ) as i!∞.
Proposition 5.2. Let K=kerφ∞ and let Gp be the isotropy subgroup of p in G.
Then
(1) φ(Γ)=φ∞(Gp)=φ∞(G) ;
(2) K acts freely on X, in particular K∩Gp={id};
(3) the short exact sequence 1!K!G−−−φ∞!φ∞(G)!1 splits with the splitting given by φ∞(G)'Gp,!G. In particular, G is a semidirect product of K and Gp.
Proof. (1) For each γ∈Γ we have d(γ(c(ti)), c(ti))!0 as i!∞, so the constant sequence γ converges to some g∈Gp. Lemma 5.1 yields φ(γ)!φ∞(g) , which means φ(γ)=φ∞(g) . Thus, φ(Γ)⊂φ∞(Gp) . Now, Gp is compact, so φ∞(Gp) is closed, and
238 i. belegradek and v. kapovitch
thereforeφ(Γ)⊂φ∞(Gp) . Sinceφ∞(Gp)⊂φ∞(G) , it remains to show thatφ∞(G)⊂φ(Γ) . Givenγ∈G, we findγi∈Γ withγi!γ. By Lemma 5.1,φ∞(γ) is the limit ofφ(γi)∈φ(Γ) , so φ∞(g)∈φ(Γ) .
(2) If k∈K fixes a point x, then 1=φ∞(k)=Pk(x)x∞ dk=Pxx∞dk=dk. Since k is an isometry, k=id .
(3) This is a formal consequence of (1) and (2).
Remark 5.3. Since G is the semidirect product of K and Gp, any γ∈G can be uniquely written as kg with k∈K and g∈Gp. We refer to k and g, respectively, as the translational part and therotational part of γ.
Remark 5.4. If Γ is discrete, then by Margulis’ lemma any finitely generated sub- group of Γ has a nilpotent subgroup whose index i and the degree of nilpotency d are bounded above by a constant depending only on n. (Of course, by [6], Γ itself is finitely generated, as is any subgroup of Γ , but we do not need this harder fact here.) The same then holds forG. Indeed, take finitely many elements gl of G and approximate them by γj,l∈Γ , so they generate a finitely generated subgroup of Γ . Then γj,li approximate gil, and, by above, gli lie in a nilpotent subgroup of Γ . Hence a d-fold iterated commutator in the gj,li ’s is trivial for all j, and then so is the corresponding commutator in the gil’s.
Hence G is nilpotent by [35, Lemma VIII.8.17].
6. Controlling injectivity radius
We continue working with the notation of §5, except now we also assume that Γ is discrete. The family (X, c(t),Γ) may have many converging subsequences with limits of the form (X∞, p, G) . We denote byK(p) theK-orbit ofp, whereK is the kernel of the rotation homomorphism G!O(n) . The goal of this section is to find a common lower bound, on the normal injectivity radii of the K(p) ’s.
Proposition 6.1. There exists a constant f(a) such that, for each x∈K(p), the norm of the second fundamental form IIx of K(p) at x is bounded above by f(a).
Proof. Since K acts by isometries, |IIx|=|IIp| for any x∈K(p) , so we can assume x=p. Let X, Y∈TpK(p) be unit tangent vectors. Extend Y to a left-invariant vector field on K(p) , and let α(t)=exp(tX)(p) be the orbit of p under the one-parameter subgroup generated by X. Since IIp(X, Y) is the normal component of ∇XY(p) , it suffices to show that |∇XY|6f(a) . Let Pp,α(t)α be the parallel transport from p to α(t) along α. By §4, we have |Pp,α(t)−Pp,α(t)∞ |6q(a)d(p, α(t))62q(a)t for all small t.
A similar argument shows that|Pp,α(t)−Pp,α(t)α |62q(a)tfor all smallt. Indeed, look at the “ruled” surface obtained by joiningpto the points ofαnear p. If we approximate
αby a piecewise geodesic curve pα(t1)... α(tk) , where α(tk)=q is some fixed point near p, then the area of the surface can be computed as the limit as k!∞ of the sum of the areas of geodesic triangles pα(ti)α(ti+1) . The area of each triangle is bounded above by d(α(ti)α(ti+1)) , so the area of the ruled surface is bounded above by the length of α from p to q, which is at most 2t, for small t.
Therefore, |Pp,α(t)∞ −Pp,α(t)α |64q(a)t=f(a)t by the triangle inequality, so
|Pp,α(t)∞ Y−Pp,α(t)α Y|6f(a)t.
On the other hand, Pp,α(t)∞ Y=Y(p) because Y is left-invariant, and since elements of K have trivial rotational parts. Thus, |Pp,α(t)α Y−Y(p)|6f(a)t, which by definition of covariant derivative implies that |∇XY(p)|6f(a) .
Corollary 6.2. (i) There exists r(a)>0 such that, if C is the connected com- ponent of K(p)∩Br(a)(p) that contains p, and if x∈Br(a)(p) is the endpoint of the geodesic segment [x, p] that is perpendicular to C at p, then d(x, c)>d(x, p) for any c∈C\{p}.
(ii) If there exists s<r(a) such that K(p)∩Bs(p) is connected, then the normal injectivity radius of K(p) is >s/3.
Proof. (i) The metric on X∞ can be approximated in C1-topology by smooth metrics with almost the same two-sided negative curvature bounds and infinite injectivity radius [28]. AlsoC1-closeness of metrics impliesC0-closeness of Levi–Civita connections, and hence almost the same bounds on the second fundamental forms of C. Now, for the smooth metrics as above the assertion of (i) is well-known, and, after choosing a slightly smaller r(a) , it passes to the limits, so we also get it for X∞.
(ii) Consider two arbitrary geodesic segments of equal length 6s/3 that start at K(p) , are normal to K(p) and have the same endpoint. Since K acts isometrically on X∞ and transitively on K(p) , we can assume that one of the segments starts at p.
By the triangle inequality, the other segment starts at a point of C, so by part (i) the segments have to coincide.
Remark 6.3. The proof that K(p)∩Bs(p) is connected, for some s<r(a) indepen- dent of the converging subsequence (X, c(ti),Γ) , occupies the rest of this section, and this is the only place in the paper where we use Bowditch’s theorem [6] that Γ is finitely generated. Other key ingredients are the existence of approximate square roots in finitely generated nilpotent groups (see Appendix A), and the following comparison lemma that relates the displacement of an element of Γ to the displacement of its square root.
240 i. belegradek and v. kapovitch
x m
g(x)
g2(x)
g(m)
v dg(v)
Figure 1.
Lemma6.4. Let U be the neighborhood of 1∈O(n)that consists of all A∈O(n) sat- isfying |Av−v|<1 for any unit vector v∈Rn. Then there exists a function f: (0,∞)! (0,∞) such that f(r)!0 as r!0 and d(g(x), x)6f(d(g2(x), x)), for any x∈X and any g∈Γ with φ(g)∈U.
Proof. Define f(r) to be the supremum of d(g(x), x) over all x∈X and g∈Γ with φ(g)∈U satisfying r=d(x, g2(x)) .
To see that f(r)<∞, take arbitrary x∈X and g∈Γ with r=d(x, g2(x)) , and let R=d(x, g(x)) . Look at the geodesic triangle in X with vertices x, g(x) and g2(x) (see Figure 1). Arguing by contradiction, assume that by choosing g and x one can make R arbitrarily large while keeping r fixed. The geodesic triangle then becomes very long and thin. Let m be the midpoint of the geodesic segment [x, g(x)] , so that g(m) is the midpoint of the geodesic segment [g(x), g2(x)] . By exponential convergence of geodesics and comparison with the hyperbolic plane of sec=−1 , we get d(m, g(m))6C(r)e−R/2, which is small since R is large. So Pg(m)m∞ is close to Pg(m)m, which in turn is close to Pg(x)mPg(m)g(x), since the geodesic triangle with vertices m, g(m) and g(x) has small area. Thus, φ(g) is close to Pg(x)mPg(m)g(x)dg. Let v be the unit vector tangent to [x, g(x)] at m and pointing towards x. Then dg(v) is tangent to [g(x), g2(x)] at g(m) and is pointing towardsg(x) . Since the geodesic triangle with verticesx, g(x) andg2(x) has small angle at g(x) , the map Pg(x)mPg(m)g(x) takes dg(v) to a vector that is close to −v. This gives a contradiction since |φ(g)(v)−v|61 .
A similar argument yields f(r)!0 as r!0 . Namely, if one can make d(g2(x), x) arbitrarily small while keeping d(g(x), x) bounded below, then the geodesic triangle with verticesx,g(x) andg2(x) becomes thin, and we get a contradiction exactly as above.
Proposition 6.5. Let r(a) be the constant of Corollary 6.2. Then there exists a positive s6r(a), depending only on X, c and Γ, such that for any converging sequence (X, c(ti),Γ)!(X∞, p, G), the normal injectivity radius of K(p) is >s.
Proof. By Corollary 6.2, it suffices to find a universals such that Ks:=K(p)∩Bs(p) is connected. Let K0s be the component of Ks containing p.
By [6] Γ is finitely generated, hence, by Margulis’ lemma [3], Γ contains a normal nilpotent subgroup Γ of indexe i6i(a, n) . Therefore H=φ(Γ) is virtually nilpotent.
Hence, its identity component H0 is abelian, since compact connected nilpotent Lie groups are abelian. Then φ−1(H0) is a subgroup of finite index in Γ . Let
Γ0=Γ∩φe −1(H0).
Clearly, [Γ:Γ0]=k=k(φ) is also finite.
We first give a proof for the case Γ=Γ0. Arguing by contradiction, suppose that for anys>0 there exists a sequence (X, c(ti),Γ)!(X∞, p, G) and a point γ(p)∈Ks\K0s with d(p, γ(p))<s. By possibly making d(p, γ(p)) smaller, we can choose γ(p) so that d(p, γ(p)) is the distance from pto Ks\K0s. By the first variation formula, the geodesic segment [p, γ(p)] is perpendicular to K0s. The next goal is to construct the square root of γ with no rotational part and displacement bounded by f(d(p, γ(p))) , where f is the function of Lemma 6.4.
Take γi∈Γ converging to γ. Since Γ is finitely generated, we can apply Lemma A.1 to find an (independent ofi) finite setF⊂Γ such that each γi can be written as γi=gi2fi
with fi∈F. We can further write eachgi as the product gi=xiri, where xi has a small rotational part and ri is close to a rotation, namely we let ri be an element of Γ that is close to φ(gi) and let xi=gir−1i . Thus γi=(xiri)2fi and
γi=xirixirifi=x2i[x−1i ri]ri2fi.
Applying Lemma A.2 to x2i[x−1i ri] we see that γi can be written as (xihi)2fi0ri2fi with hi∈[Γ,Γ] and fi0∈F0. Sinceφ(Γ) is abelian, we have φ(hi)=1 and hence φ(xihi)=φ(xi) is small. Since F and F0 are independent of i, each element of F and of F0 is close to a rotation for large i. So fi0ri2fi is close to a rotation and since both φ(γi) and φ(xihi) are small,fi0ri2fi subconverges to the identity. Thus, (xihi)2 subconverges to γ, and we might as well assume that γi=(xihi)2 in the beginning. By Lemma 6.4,
d(c(ti), xihi(c(ti)))6f(d(c(ti),(xihi)2(c(ti)))),
where the right-hand side converges to f(d(p, γ(p))) . Hence xihi subconverges to w∈K such that w2=γ and d(p, w(p))6f(d(p, γ(p))) .
242 i. belegradek and v. kapovitch
Since w has no rotational part, Ppw(p)∞ =dwp. By assumption, s can be taken arbitrarily small, so we can assume that f(d(p, γ(p))) is small, in particular w(p)∈Ks. So Ppw(p) is close to dwp. Hence, if v is the unit vector tangent to [p, w(p)] at p and pointing towards w(p) , then Ppw(p)(v) is close to dwp(v) . Therefore, w(p) is close to the midpoint of [p, w2(p)]=[p, γ(p)] . Since [p, γ(p)] is perpendicular to K0s and d(p, γ(p))<r(a) , it is clear that w(p)∈K/ 0s. This contradicts the minimality of d(p, γ(p)) and completes the proof in the case Γ=Γ0.
We now turn to the general case. Let G0 be the subset of G that consists of limits of elements of Γ0 under the convergence (X, c(ti),Γ)!(X∞, p, G) . It is straightforward to check that G0 is a closed subgroup of G of index 6k. Thus the limit of any con- verging subsequence of (X, c(ti),Γ0) has to be equal to (X∞, p, G0) , therefore, in fact, (X, c(ti),Γ0) converges to (X∞, p, G0) . Since the rotation homomorphism of G restricts to the rotation homomorphism of G0, the translational part K0 of G0 is G0∩K. In par- ticular,|K:K0|6k, hence the identity components ofK andK0 coincide. Using the first part of the proof, we fixssuch thatK0(p)∩Bs(p) is connected. ThusK0(p)∩Bs(p)=K0s. Now let γ(p)∈Ks\K0s be such that d(p, γ(p)) is the distance from p to Ks\K0s. Then the geodesic segment [p, γ(p)] is perpendicular toK0s by the first variation formula.
Arguing by contradiction, suppose that d(p, γ(p)) can be arbitrarily small. Then, by the triangle inequality,γj(p) is close top for j=1, ..., k. Also, γk∈K0, so in fact γk(p)∈K0s because K0(p)∩Bs(p)=K0s.
On the other hand, since the γj’s have no rotational part, the argument used above to prove that w(p) is close to the midpoint of [p, w2(p)] shows that the points γj(p) almost lie on a geodesic segment [p, γk(p)] . Then the segments [p, γk(p)] and [p, γ(p)]
have almost the same direction, so [p, γk(p)] is almost perpendicular to K0s. Hence by Corollary 6.2, ifd(p, γ(p)) is small enough, thenγk(p)∈K/ 0s, which is a contradiction.
Remark 6.6. Although it is not needed for the proof of Theorem 1.1, note that all possible limits of (X/Γ, σ(ti)) have the same dimension independent of the sequence ti!∞. Consider all possible limits with fixed dim(K) , and look at the space X∞/K. It has a lower bound on the injectivity radius for points near the projection ¯pof p, and hence vol(B(¯p,1))>c>0 in all such spaces, where c is independent of the converging sequence. By Proposition 5.2, the isotropy group Gp is the same for all possible limits and, moreover, the Gp-actions on TpX∞ are all equivalent. Also, by Lemma A.3, the identity component Gidp of Gp commutes with the identity component of K, hence Gidp fixes pointwise the component ofK(p) containingp. Thus the Gidp -actions onTp¯X∞/K are all equivalent. This implies that a unit ball in X∞/G has volume >c0>0 with c0 only depending on dim(K) . Hence all limits of the same dimension form a closed subset among all limits; therefore, the space of all limits is the union of these closed sets. On
the other hand, the space of all limits is connected by Lemma 6.7 below, thus all the limits have the same dimension.
Lemma6.7. If γ: [0,∞)!Z is a continuous precompact curve in a metric space Z, then the space Lim(γ) of all possible subsequential limits limti!∞γ(ti) is connected.
Proof. If Lim(γ) is not connected, then we can write it as a disjoint union of closed (and hence compact) sets Lim(γ)=AtB. ThenUε(A)∩Uε(B)=∅for some ε>0 , where Uε(S) denotes the ε-neighborhood of S. Let γ(ti)!a∈A and γ(t0i)!b∈B. Arguing by contradiction, we see that the curve γ|[ti,t0i] lies in Uε(Lim(γ))=Uε(A)∪Uε(B) for all large i. Clearly, γ(ti)∈Uε(A) and γ(t0i)∈Uε(B) for all large i, which contradicts Uε(A)∩Uε(B)=∅.
7. Product structure at infinity
In the next two sections we apply the critical point theory for distance functions to show the following result.
Theorem 7.1. For each large t, the horosphere quotient Ht/Γ is diffeomorphic to the normal bundle of an orbit of an N-structure on Ht/Γ.
Proof. Let σ(t) be the projection of c(t) to X/Γ . Given a converging sequence (X, c(ti),Γ)!(X∞, p, G) , the sequence of pointed Riemannian manifolds (X/Γ, σ(ti)) converges in pointed Gromov–Hausdorff topology to a pointed Alexandrov space (Y, q):=
(X∞/G, q) with curvature bounded below by −a2.
The identity component Kid of K is normal in K, hence its p-orbit Kid(p) is invariant under the action of Gp. Let rs be a positive constant to be determined later, where s comes from Proposition 6.5. The 3r-tubular neighborhood of Kid(p) is also Gp-invariant, so the ball B3r(q) is isometric to the G-quotient of this tubular neighborhood. By Proposition 6.5, anyx∈B3r(q) can be joined toq by a unique shortest geodesic segment [q, x]⊂B3r(q) .
Recall that in general a distance function d(·, q) on an Alexandrov space is called regular at the point x if there exists a segment emanating from x that forms an angle
>π/2 with any shortest segment joining x to q.
In our case, the function d(·, q) is regular at any x∈B3r(q)\{q}.
Let w(a)>1 be a constant depending only on a that will be specified later. By angle comparison, the function d(·, σ(t)) on
Ar(σ(t)) ={x∈Bs(σ(t)) :d(x, σ(t))∈[r/w(a), w(a)r]}
244 i. belegradek and v. kapovitch
is regular provided the Gromov–Hausdorff distance between Bs(q) and Bs(σ(t)) is r/w(a) . Because the family{Bs(σ(t))} is precompact in the Gromov–Hausdorff topol- ogy, Proposition 6.5 implies that the function d(·, q) is regular on Ar(σ(t)) for all t>t0
with sufficiently large t0.
We denote by Ht the horosphere centered at c(∞) that contains c(t) . Since the second fundamental form of Ht is bounded in terms of a, any short segment joining nearby points of Ht/Γ is almost tangent to Ht/Γ . Hence, by taking r sufficiently small, we can assume that for all t>t0 and all x∈Ar(σ(t)) there exists a unit vector λ∈Tx(Ht/Γ) that forms an angleαλ,[x,σ(t)]∈2
3π, π
with any shortest segment [x, σ(t)] . By the first variation formula, the derivative of d(·, q) in the direction of λ equals the minimum of −cosαλ,[x,σ(t)], over all shortest segments [x, σ(t)] , and by above it lies in 1
2,1 .
The distance function on X/Γ need not be smooth, and for what follows it is con- venient to replace d(·, σ(t)) by its average over a small ball Bδ(σ(t)) as follows. Given δr, define f:X/Γ!R by
f(x) = 1 volBδ(σ(t))
Z
Bδ(σ(t))
d(x, y)dy,
where x∈Ht/Γ (i.e. t=b(x) ). Now, f is a C1 1 -Lipschitz function with
|f(x)−d(x, σ(t))|6δ for any x∈Ar(σ(t)) . Observe that, for any η∈Tx(X/Γ) ,
dfx(η) = 1 volBδ(σ(t))
Z
Bδ(σ(t))
(−cosαη,[x,y])dy. (7.2)
Also note that, since δr and sec(X/Γ)>−a2, for all large t, if x∈Ar(σ(t)) and y∈Bδ(σ(t)) , then there is a point z such that d(z, x)≈d(x, y) and d(z, y)≈2d(x, y) . Therefore, the angle corresponding to x in the comparison triangle in the space of sec≡−a2 is almost π. By Toponogov comparison, the angle at xin any geodesic triangle 4xyz is almost π. By (7.2), this implies that if η is a direction of any shortest segment connecting x to z, then dfx(η)∈1
2,1
providedδ is small enough.
By gluing λ’s via a partition of unity, we obtain a C1 unit vector field Λ that is tangent to Ht/Γ and defined for all t>t0 and x∈Ar(σ(t)) , and such that dfx(Λ)∈1
4,1 if δ is sufficiently small. Then, Sr={x∈X/Γ:f(x)=r} is a properly embedded C1- hypersurface in X/Γ that is transverse to Λ . Also, the compact submanifold St(t):=
Sr∩Ht/Γ is δ-close to the metric r-sphere in Ht/Γ centered at σ(t) . Furthermore, A(r, t):=Ar(σ(t))∩Ht/Γ is C1-diffeomorphic to the product Sr(t)×[r/w(a), w(a)r] .
Here we are only interested in the part of X/Γ with t>t0. There, the Busemann functionX/Γ!R restricts to a C1-submersion Sr![t0,∞) , because otherwise at some point the tangent spaces of Sr and Ht/Γ coincide by dimension reasons, so that Sr
cannot be transverse to Λ∈T(Ht/Γ) . By construction, the submersion is proper, hence it is a C1-fiber bundle, which is C1-trivial by the covering homotopy theorem. The trivialization defines a C1-isotopy F:Sr(t0)×[t0,∞)!X/Γ such that Sr(t0)×{t} is mapped onto Sr(t) .
We push this isotopy along the Busemann flow back into Ht0/Γ by setting
G(t, x) =bt0−t(F(x, t))
for any t>t0 and x∈Sr(t0) , to get the C1-isotopy G:Sr(t0)×[t0,∞)!Ht0/Γ .
The Busemann flow induces a C1-diffeomorphism Ht/Γ!Ht0/Γ , so around each submanifold bt0−t(Sr(t)) there is a “tubular neighborhood” bt0−t(A(r, t)) .
By the exponential convergence of geodesics, one can choose w(a) in the definition of Ar(σ(t)) so that, for any t>t0, there exists t0>t+1 such that bt0−t0(Sr(t0)) is contained in bt0−t(A(r, t)) and is disjoint from bt0−t(Sr(t)) . By the following elementary lemma, the region between bt0−t(Sr(t)) and bt0−t0(Sr(t0)) is C1-diffeomorphic to Sr(t)×[0,1] .
Lemma 7.3. Let M be a closed smooth manifold and Ft:M!M×R be a C1- isotopy with F0(M)=M×{0}. If F0(M) and Fs(M) are disjoint for some s, then the region between F0(M) and Fs(M) is diffeomorphic to M×[0,1].
Proof. By the isotopy extension theorem [12, p. 293], we can extend the isotopy Ft to an ambient C1-isotopy which is the identity outside a compact subset of M×R.
Assume, without loss of generality, that s<0 , and then take n>0 so large that the isotopy is the identity on M×{n}. Then, by restricting the ambient isotopy to the region between M×{n} and M×{0}, we get a diffeomorphism of the region between M×{n} andM×{0} onto the region betweenM×{n} andFs(M) . The former region is the product, so is the latter. But the latter region is diffeomorphic to the region between M×{0} andFs(M) , because the region between M×{n} andM×{0} isM×[0,1] .
By gluing a countable number of such diffeomorphisms together we conclude that, for all sufficiently large t,
(Ht/Γ)\U(r, t) is C1-diffeomorphic to [t,∞)×Sr(t), (7.4)
where U(r, t)={x∈Ht/Γ:f(x)<r}.
246 i. belegradek and v. kapovitch
8. Tubular neighborhood of an orbit
It remains to understand the topology of U(r, t) , and we do so for large enough t and small enough r. The proof involves the collapsing theory developed in [14] and the geometry of Alexandrov spaces (for which we refer to [9] and Appendix D).
Let us look at a converging sequence (Hti/Γ, σ(ti))!(H, q) . First, we replace the metric on Hti/Γ with an invariant Riemannian metric which is εi close to Hti/Γ in C1-topology and is A(εi) -regular [37], [14], [28], where εi!0 as i!∞. Also, the spaces Hti/Γ with the new metrics have uniform curvature bound |sec|6C0 [37] that depends only on the original curvature bound of Hti/Γ . The collapsing theory [14] yields, for each i, the following commutative diagram given by the invariant metric hi on Hti/Γ :
F Bi
ηi //Yi
Bi ηi //Xi.
Here Bi is the ball B(σ(ti),1) in the metric hi, and F Bi is the frame bundle ofBi. The vertical arrows are quotient maps under isometric O(n) -actions and ηi is a Riemannian submersion given by the N-structure on F Bi. Clearly the induced map ηi is a subme- try (see Appendix D for background on submetries). By Lemma D.3, the Toponogov comparison with curv>−C0 holds for any triangle with vertices in B ηi(σ(ti)),14
for all large i.
Since we will only be interested in the geometry of Xi inside the 18-neighborhood of ηi(σ(ti)) , we will treat the Xi’s as Alexandrov spaces.
Note that Xi
−−−G–H!X=B(q,1) and dimXi=dimX for all large i. We claim that there exists an R>0 and a sequence qi∈Xi converging to q such that d(·, qi) has no critical points in B(qi, R) for all large i. Consider two cases depending on whether q lies on the boundary of the Alexandrov space X.
Case 1. Suppose q /∈∂X. Since X has curv>−C0 in comparison sense, by [21], there exists a strictly concave function uon a B(q, R) , for some R1 , such that it has a maximum at q, and the superlevel sets are compact. By possibly making R smaller, we can assume that R<d(q, ∂X) . This function is constructed by taking averages and minima of distance functions. Therefore, it naturally lifts to a function ui on Xi such that ui converges uniformly to u. By [21, Lemma 4.2], the liftsui are strictly concave on B(ηi(σ(ti)), R/2) for all large i. Let qi be the point of maximum of ui. By uniqueness of the maximum, qi!q as i!∞. By Lemma D.1, d(·, qi) has no critical points in B(qi, R/3) for all large i.
Case 2. Suppose now that q∈∂X. Denote by DXand DXi the doubles of X and Xi along the boundary and let ιbe the canonical involution. By [30], the doubles are also Alexandrov spaces with curv>−C0. It is clear that DXi
−−−G–H!DX. By construction, we can choose u and ui to be ι-invariant. As before, let qi be the point of maximum of ui. Since ui(qi)=ui(ι(qi)) , by uniqueness of maxima of strictly concave functions, we see that qi must lie on ∂Xi. Again by Lemma D.1, d(·, qi) has no critical points in B(qi, R/3) in DXi for all large i, and hence the same is true for d(·, qi) in B(qi, R/3) . This immediately implies that the distance function d(·, Oi) to the orbitOi over qi
has no critical points in theR/3 -neighborhoodUR/3(Oi) of Oi for all largei. Indeed, let x∈UR/3(Oi)\Oi and letγ(t) be a geodesic starting atηi(x) such that dtdd(γ(·), qi)0|t=0>
0 . Sinceηi: (H(ti)/Γ, hi)!Xi is a submetry, there exists a horizontal lifteγ of γ starting at x. Then d(eγ(t), Oi)=d(γ(t), qi) , and hence d(eγ(t), Oi)0|t=0=d(γ(t), qi)0|t=0>0 .
Therefore Ur(Oi) is diffeomorphic to the total space of the normal bundle toOi in Hti/Γ for any r6R/3 . Since Oi is Hausdorff close to σ(ti) , the same is true forU(r, ti) . Combining this with (7.4), we conclude that Hti/Γ is diffeomorphic to the total space of the normal bundle to Oi in Hti/Γ for all sufficiently large i. Finally, since the above proof works for any sequence ti!∞, arguing by contradiction we conclude that Ht/Γ is diffeomorphic to the normal bundle of an orbit of an N-structure for all sufficiently large t. This completes the proof of Theorem 7.1.
Remark 8.1. The reader may be wondering why we work with the Alexandrov spaces Xi instead of the Riemannian manifolds Yi. This is because the curvature of Yi may tend to±∞as i!∞, which makes it hard to control the geometry of theYi’s. If instead ofεi!0 , we take εi equal to a small positive constant ε, then |sec(Yi)|6C(ε) ; but then it may happen that the injectivity radius of the Gromov–Hausdorff limit of the Yi’s is ε, so we cannot translate the lower bound on the injectivity radius from Yi to Hti/Γ . Remark 8.2. By Theorem 7.1, each orbit Oqi as above is homotopy equivalent to X/Γ . Thus all the Oqi’s are homotopy equivalent, hence they are all affinely diffeomor- phic (see e.g. [39, Theorem 2]).
9. The normal bundle is flat
Theorem 9.1. For each large t, the horosphere quotient Ht/Γ admits an N- structure that has an orbit Ot such that the normal bundle to Ot is a flat Euclidean vector bundle with total space diffeomorphic to Ht/Γ.
248 i. belegradek and v. kapovitch
Arguing by contradiction, it suffices to prove the theorem for any sequence ti!
∞ such that Hti/Γ converges in pointed Gromov–Hausdorff topology. We fix such a sequence and assume for the rest of the proof that t belongs to the sequence.
We denote by gt the C1-Riemannian metric on the horosphere quotient Ht/Γ in- duced by the ambient metric (M, g) . Fix a small positive ε>0 to be determined later;
this constant will only depend on (M, g) . By Theorem 7.1, [14] and [28], for all large t there exists an N-structure on Ht/Γ with an orbit Ot such that the normal bundle to Ot is diffeomorphic to Ht/Γ . Also all orbits of the N-structure have diameter <ε with respect to an invariant metric ht that is ε-close to gt in uniform C1-topology, i.e.
|gt−ht|<ε and|∇gt−∇ht|<ε. It remains to show that for all larget, the normal bundle to Ot in Ht/Γ is flat Euclidean. The proof breaks into two independent parts.
In§9.1 we find a stratum Ft of the N-structure on Ht/Γ such that Ft is an ht- totally-geodesic closed submanifold that contains Ot with flat normal bundle. This uses only general properties of N-structures.
In§9.2 we show that the restriction to Ot of the normal bundle of Ft in Ht/Γ is flat, for all large t. This uses the flat connection of§4 and the fact that Ot!Ht/Γ is a homotopy equivalence.
9.1. The normal bundle in a stratum is flat
Throughout§9.1 we suppress the index t, writing O in place of Ot, etc. Let V be the tubular neighborhood ofO that is sufficiently small so that all orbits inV have dimension
>dim(O) . Let Oe and Ve be their universal covers. According to [14, pp. 364–365], the group Iso(Ve) contains a connected (but not necessarily simply-connected) nilpotent subgroup N that stabilizes Oe, acts transitively on Oe, and also Λ=N∩π1(V) is a finite index subgroup of π1(V) and is a lattice in N. The following lemma is implicit in [14].
Lemma 9.2. The subgroup H of Iso(Ve) generated by N and π1(V) is closed, N is the identity component in H, and the index of N in H is finite.
Proof. Λ is a cocompact discrete subgroup ofN and also of its closure Nin Iso(Ve) . Since dim(N) and dim(N) are both equal to the cohomological dimension of Λ , we get N=N. Now let Λ0 be a maximal finite indexnormalsubgroup ofπ1(Ve) that is contained in Λ . Ifγ∈π1(V) , thenN∩γN γ−1 contains Λ0 as a cocompact discrete subgroup, thus, as before, dim(N) and dim(N∩γN γ−1) are both equal to the cohomological dimension of Λ0, so N=N∩γN γ−1, and N is normalized by π1(V) . Thus N is normal in H, and Λ=Λ0. Since N is connected, it remains to show that |H:N| is finite. Since N and π1(V) generate H, the finite subgroup π1(V)/Λ of H/N generates H/N, hence π1(V)/Λ=H/N.
Let I be the intersection of the isotropy subgroups of H of the points of Oe. Since Oe is H-invariant, I is normal in H. The fixed point set of I is a totally geodesic submanifold Fe of Ve. The H-action on Fe descends to an H/I-action on Fe. Since π1(V) is torsion free and discrete, π1(V)∩I is trivial, and we identify π1(V) with its image in H/I. Denote the projection of Fe into V by F.
Lemma 9.3. The normal bundle to O in F is flat.
Proof. The group N acts transitively on Oe, so all isotropy subgroups for the N- action on Oe are conjugate. Since they are also compact, they lie in the center of N [20];
in particular, all the isotropy subgroups are equal, and hence each of them is equal to I∩N. In particular, N/(I∩N) acts freely and transitively on Oe. Since Oe is simply- connected, so isN/(I∩N) . Thus, I∩N is the maximal compact subgroup of N; hence, by Lemma A.3, I∩N is a torus, which we denote by T. The torus is the identity component of the compact group I, because |I:I∩N|6|H:N|<∞. Since N/T acts freely and transitively on Oe, we can choose a trivialization of ν, the normal bundle of F in Ve that is invariant under the left translations by N/T. Namely, let e∈Oe be the point corresponding to 1∈N/T under the diffeomorphism N/T∼=Oe. Fix an isomorphism φ: νe!{e}×Rk, and then extend it to the N/T-left-invariant isomorphism ν∼=O×Re k. Now, take γ∈π1(V) and x∈Oe. Using the above trivialization we define the rotational part of γ as the automorphism of {e}×Rk given by
φdLγ(x)−1dγdLxφ−1,
wheredLxis the differential of the left translation byx∈N/T. SinceO is an infranilman- ifold, π1(V) acts on Oe by affine transformations, that is if y∈Oe, then γ(y)=nγ·Aγ(y) , where nγ∈N/T and Aγ is a Lie group automorphism of N/T. Hence, for z∈Oe, we get
(Lγ(x)−1γLx)(z) =γ(x)−1·γ(xz) =Aγ(x)−1·n−1γ ·nγ·Aγ(xz) =Aγ(z) =Ln−1 γ γ(z), where the third equality holds as A is an automorphism and N/T=Oe. This establishes the above equality only on Oe, not on Fe, but both sides of the equality make sense as elements of H/I, and since Fe is the fixed point set of I, any two elements of H/I that coincide on Oe must coincide on Fe. Now the right-hand side is independent of x, which implies that the rotational part of γ is independent of x. This means that the bundle (Oe×Rk)/π1(V) is a flat O(k) -bundle, and hence so is the normal bundle ofF in V.
9.2. The normal bundle to a stratum is flat Let Ft be the stratum from§9.1.
